An eigenvalue interlacing approach to Garland's method

TL;DR

This paper extends Garland's vanishing theorem for simplicial complexes by using an eigenvalue interlacing approach to relate homology dimensions to spectral properties of links, providing a new local-to-global principle.

Contribution

It introduces a novel eigenvalue interlacing method to generalize Garland's theorem, connecting spectral data of links to homology vanishing conditions.

Findings

Generalizes Garland's vanishing theorem using eigenvalue interlacing

Provides bounds on homology dimensions based on spectral data

Introduces an abstract local-to-global principle for simplicial complexes

Abstract

Let $X$ be a pure $d$-dimensional simplicial complex. For $0\le k\le d$, let $X(k)$ be the set of $k$-dimensional faces of $X$, let $\tilde{L}_k(X)$ be the $k$-dimensional weighted total Laplacian operator on $X$, and let $\tilde{H}_k(X;\mathbb{R})$ be its $k$-dimensional reduced homology group with real coefficients. For $\sigma\in X$, let $\text{lk}(X,\sigma)$ be the link of $\sigma$ in $X$. For a matrix $M$, we denote by $\text{Spec}(M)$ the multi-set containing all the eigenvalues of $M$. We show that, for every $0\le \ell<k \le d$, \[ \text{dim}(\tilde{H}_k(X;\mathbb{R}))\le \sum_{\eta\in X(\ell)}\left| \left\{ \lambda\in \text{Spec}(\tilde{L}_{k-\ell-1}(\text{lk}(X,\eta))) :\, \lambda\le \frac{(\ell+1)(d-k)}{k+1}\right\}\right|. \] This extends the classical vanishing theorem of Garland, corresponding to the special case when the right hand side of the inequality is equal to zero, and a more recent result by Hino and Kanazawa, corresponding to the case $\ell=k-1$. A main new ingredient in our proof is an abstract version of Garland's local to global principle, which follows as a simple consequence of the eigenvalue interlacing theorem, and may be of independent interest.